Overlimiting Current in a Microchannel

We revisit the classical problem of diffusion-limited ion transport to a membrane (or electrode) by considering the effects of charged sidewalls. Using simple mathematical models and numerical simulations, we identify three basic mechanisms for overlimiting current in a microchannel: (i) surface conduction carried by excess counterions, which dominates for very thin channels, (ii) convection by electro-osmotic flow on the sidewalls, which dominates for thicker channels, and (iii) transitions to electro-osmotic instability on the membrane end in very thick channels. These intriguing electrokinetic phenomena may find applications in biological separations, water desalination, and electrochemical energy storage.

Figure 1

Physical picture of overlimiting current (OLC) in a microchannel from a reservoir (left) to a cation-exchange membrane (right). The volume-averaged conductivity profile (a) exhibits a classical linear diffusion layer (continuous line) and a constant depleted region (dashed line), where the current is carried primarily by (b) SC, (c) EOF, or (d) EOI with increasing channel thickness.

Figure 2

Overlimiting current in the surface conduction regime: (a) Dimensionless current-voltage relation, Eq. (4), for ${\tilde{\rho }}_s=-0.01$, $-0.1$ and (b) mean concentration profile of inert anions, Eq. (3), for ${\tilde{\rho }}_s=-0.1$. (c) The equivalent circuit: an ideal diode for bulk CP in parallel with a constant shunt resistance for SC (or EOF, as in Fig. 3).

Figure 3

2D simulations of the EOF mechanism for OLC (${\tilde{\lambda }}_0=0.01$, $\tilde{H}=0.2$, ${\tilde{\sigma }}_s=-0.01$): (a) bulk salt concentration $\tilde{c}$ (contours) and velocity vectors $\mathbf{u}$ (arrows); (b) area-averaged concentration profiles (continuous), showing a nearly constant depleted-zone concentration ${\tilde{c}}_d\approx {\tilde{l}}_c$. Inset: vortex position ${\tilde{l}}_c$ vs voltage $\tilde{V}$; (c) current-voltage relation from simulations (continuous) and the 1D model (8) (dashed). Inset: area-averaged concentration profile in the 2D simulations (continuous) and in 1D model (dashed) for $\tilde{V}=50$.

Figure 4

The regions of dominance of the SC, EOF, and EOI mechanisms for varying channel aspect ratio, ${\tilde{H}}^{-1}$ and surface charge ${\tilde{\sigma }}_s$ for ${\tilde{\lambda }}_0={10}^{-5}$. The critical value of $\tilde{H}$ is given by $\Sigma =1$ in (10) for the SC/EOF transition and by $\frac{d{\tilde{c}}_d}{d\tilde{H}}=0$ in 2D simulations for the EOF/EOI transition. Inset: Simulated vortex center location ${\tilde{l}}_c$ versus $\tilde{H}$ for ${\tilde{\lambda }}_0=0.01$, ${\tilde{\sigma }}_s=-0.1$ (solid) compared to the scaling law, ${\tilde{l}}_c=0.38{\tilde{H}}^{4/5}$ in (9).